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arXiv · 2504.17589

MacWilliams Theory over Zk and nu-functions over Lattices

Zhiyong Zheng·Fengxia Liu·Kun Tian
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Abstract

Continuing previous works on MacWilliams theory over codes and lattices, a generalization of the MacWilliams theory over $\mathbb{Z}_k$ for $m$ codes is established, and the complete weight enumerator MacWilliams identity also holds for codes over the finitely generated rings $\mathbb{Z}_k[\xi]$. In the context of lattices, the analogy of the MacWilliams identity associated with nu-function was conjectured by Sol\'{e} in 1995, and we present a new formula for nu-function over the lattices associated with a ternary code, which is rather different from the original conjecture. Furthermore, we provide many counterexamples to show that the Sol\'{e} conjecture never holds in the general case, except for the lattices associated with a binary code.

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Zhiyong Zheng, Fengxia Liu, Kun Tian. 2025-04-24. MacWilliams Theory over Zk and nu-functions over Lattices. https://arxiv.org/abs/2504.17589

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